A026800 Number of partitions of n in which the least part is 7.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 18, 20, 24, 27, 32, 36, 42, 48, 56, 63, 73, 83, 96, 108, 125, 141, 162, 183, 209, 236, 270, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153, 1291, 1453, 1628, 1829, 2045

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

From Jason Kimberley, Feb 03 2011: (Start)

a(n) is also the number of not necessarily connected 2-regular graphs on n-vertices with girth exactly 7 (all such graphs are simple). The integer i corresponds to the i-cycle; the addition of integers corresponds to the disconnected union of cycles.

By removing a single part of size 7, an A026800 partition of n becomes an A185327 partition of n - 7. (End)

Examples

			a(0)=0 because there does not exist a least part of the empty partition.
The  a(7)=1 partition is 7.
The a(14)=1 partition is 7+7.
The a(15)=1 partition is 7+8.
.............................
The a(20)=1 partition is 7+13.
The a(21)=2 partitions are 7+7+7 and 7+14.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth exactly g

Crossrefs

Cf. A185327 (Mathematica code)

Not necessarily connected 2-regular graphs with girth at least g [partitions into parts >= g]: A026807 (triangle); chosen g: A000041 (g=1 -- multigraphs with loops allowed), A002865 (g=2 -- multigraphs with loops forbidden), A008483 (g=3), A008484 (g=4), A185325(g=5), A185326 (g=6), A185327 (g=7), A185328 (g=8), A185329 (g=9).

Not necessarily connected 2-regular graphs with girth exactly g [partitions with smallest part g]: A026794 (triangle); chosen g: A002865 (g=2 -- multigraphs with at least one pair of parallel edges, but loops forbidden), A026796 (g=3), A026797 (g=4), A026798 (g=5), A026799 (g=6), this sequence (g=7), A026801 (g=8), A026802 (g=9), A026803 (g=10). - Jason Kimberley, Feb 03 2011

Programs

Magma

p :=  func< n | n lt 0 select 0 else NumberOfPartitions(n) >;
A026800 := func< n | p(n-7)-p(n-8)-p(n-9)+p(n-12)+2*p(n-14)-p(n-16)- p(n-17)-p(n-18)-p(n-19)+2*p(n-21)+p(n-23)-p(n-26)-p(n-27)+p(n-28) >; // Jason Kimberley, Feb 03 2011

Magma

R:=PowerSeriesRing(Integers(), 75); [0,0,0,0,0,0,0] cat Coefficients(R!( x^7/(&*[1-x^(m+7): m in [0..80]]) )); // G. C. Greubel, Nov 03 2019

Maple

N:= 100: # for a(0)..a(N)
S:= series(x^7/mul(1-x^i,i=7..N-7),x,N+1):
seq(coeff(S,x,i),i=0..N); # Robert Israel, Jul 04 2019

Mathematica

CoefficientList[Series[x^7/QPochhammer[x^7, x], {x, 0, 75}], x] (* G. C. Greubel, Nov 03 2019 *)
Join[{0},Table[Count[IntegerPartitions[n],?(#[[-1]]==7&)],{n,80}]] (* _Harvey P. Dale, Apr 05 2025 *)

PARI

my(x='x+O('x^75)); concat([0,0,0,0,0,0,0], Vec(x^7/prod(m=0,80, 1-x^(m+7)))) \\ G. C. Greubel, Nov 03 2019

Sage

def A026800_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^7/product((1-x^(m+7)) for m in (0..80)) ).list()
A026800_list(75) # G. C. Greubel, Nov 03 2019

Formula

G.f.: x^7 * Product_{m>=7} 1/(1-x^m).

a(n) = p(n-7) -p(n-8) -p(n-9) +p(n-12) +2*p(n-14) -p(n-16) -p(n-17) -p(n-18) -p(n-19) +2*p(n-21) +p(n-23) -p(n-26) -p(n-27) +p(n-28) where p(n)=A000041(n) including the implicit p(n)=0 for negative n. - Shanzhen Gao, Oct 28 2010; offset corrected / made explicit by Jason Kimberley, Feb 03 2011

a(n) ~ exp(Pi*sqrt(2*n/3)) * 5*Pi^6 / (6*sqrt(3)*n^4). - Vaclav Kotesovec, Jun 02 2018

G.f.: Sum_{k>=1} x^(7*k) / Product_{j=1..k-1} (1 - x^j). - Ilya Gutkovskiy, Nov 25 2020

Extensions

More terms from Arlin Anderson (starship1(AT)gmail.com), Apr 12 2001